Generalized Free Wave Transfer Matrix Method for Solving the Schrödinger Equation With an Arbitrary Potential Profile

Transfer matrix method has been widely used to investigate the quantum phenomena in semiconductor materials and optoelectronic devices with an arbitrary potential profile. Free wave or complex exponential form transfer matrix method is easy to implement and also proved to be an efficient and accurate approach. However, there are numerical singularities in these kinds of matrices when the electron energy is equal to the potentials of certain segments. The singularity greatly limits the applications of this method. In this paper, by investigating the local zero energy problem of the Schrodinger equation, a generalized free wave transfer matrix method is formulated without any singular point in all possible electron energy range. Its validity and accuracy are shown both mathematically and numerically. The application is demonstrated by an example related to quantum well intermixing effect.